On his return he removed to Berlin, where he lived as a royal pensioner till his death, which occurred on the 18th of February 18 His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta-function, as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Konigsberg, 1829), and in later papers in Crelle's Journal, constitute his grandest analytical discoveries.

Hempl's initiative was followed by Professor Gundermann of Giessen, who announced in November 1897' that he had discovered the source of the runic alphabet, the introduction of which he declares preceded the first of the phonetic changes known as the " Teutonic sound-shifting," since < = g is used for k, X = x for g, a Theta-like symbol for d, while zd is used for st.

If we multiply theta by p, q, r, respectively, or again by Ap, Bq, Cr respectively, and add, we verify that the expressions Ap2 + Bqf + Cr1 and A1p2 + Bfqi + Ciri are both consta~it.